
About A215940, A217626, A185310, A188965, and A196020

Conclusive observations:
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This sequence (A185310) actually doesn't exist.

It is simply another alternative way for representing the terms in A215940.

What currently is part of the b-file for A215940 actually is not such sequence.

What actually happens there is that we had taken a set of column vectors containing
the digits of a permutation (Every natural number is a digit relative to an enough big radix )
as its components, evaluating a polynomial and working with these results.

The true nature of A215940 is not ONLY based on the elementary arithmetic supported by the database.
There might seem now unnecessary the polynomial evaluation and the divisions by (base -1), but those
operations actually are equivalent to this more general procedure:

1) Take a set of the first N! permutations for the first N natural numbers and arrange them in ascending order.

2) After done (1), store them as column vectors.

3) Pick each one on those column vectors and subtract from them the first.

4) Pick each one of the column vector differences obtained in (3) and operate with the unitriangular non-strict
upper matrix (multiply by the left the described matrix by the column vector). Do all the mentioned multiplied by (-1).

Note:  In PARI/GP, such matrix is easily defined as for example

? A(n)=matrix(n,n,i,j,(i<=j));

Then for example of the steps described above compute in PARI/GP:

? A(n)=matrix(n,n,i,j,(i<=j));
? -1*A(11)*([11,10,9,8,7,6,5,4,3,2,1]~-[1,2,3,4,5,6,7,8,9,10,11]~)

And excluding the first component (always a zero) what remains is a palindromic  pattern.  As I commented in A215940,
this is a property for the greatest and the smallest possible permutations for N naturals (I said there the last term
among the first N! instead but it means the same). the greatest component value of the result gives the lower bound
for the radices where the terms of A215940 computed in a base-specific way looks invariant. 

This is it. The true universality of A215940 lies on the tensors and not in the integer numbers subject to a specific
base. The same happens to A217626.

A185310 is just another way to sort the differences among the column vector representations described previously. When
everything is turned into polynomials by setting a radix, but sequences looks quite different.

Please take a look to the divisibility proof linked to A215940. There is more clear the connection between the
transformation matrix described and the polynomial remainder theorem stated in a Diophantine context.

Also additionally, a nice identity relative to the binomial coefficients was found in connection with this work.

R. J.  Cano, Feb 1 2013. 08:01 VET